
Furthermore, by the standard Hölder estimate (see [1]}),
the iterative estimate () and (REF ), for \(1\le N\le 4\) ,
\(& \Vert  \rho _B \Vert _{C_{t,x}} \lesssim \ell ^{-1} , \quad \Vert  \rho _B \Vert _{C_{t,x}^N} \lesssim \ell ^{-N}\delta ^{-N+1}_{q+1}, \\&\Vert  \rho _B^{1/2}\Vert _{C_{t,x}} \lesssim \ell ^{-1}, \quad \Vert  \rho _B^{1/2} \Vert _{C_{t,x}^N} \lesssim \ell ^{-N} \delta ^{-2N}_{q+1},  \\&\Vert  \rho _B^{-1}\Vert _{C_{t,x}}\lesssim \delta _{q+ 1}^{-1} ,\quad \Vert  \rho _B^{-1} \Vert _{C_{t,x}^N } \lesssim \ell ^{-N}\delta ^{-2N}_{q+1} . \) 
